Consider a bird whose flying to the warm regions before the arrival of winter and since they have to fly for longer distances and time their objective it to maintain certain height. Now if we make a simple model of height maintained by the bird we can write it as:
Height of the bird maintained = f (no of times wings flapped)
So in order to maintain certain height, certain number of wings needed to be flapped, if bird flaps less it will reduce altitude and it will risk being hunted and if it flaps more it will gain more altitude and risk its life because of less oxygen or extra cold.
Here as both variables can change naturally if there are no targets mean bird is randomly flying from one place to another. Since there are targets (maintaining certain heights) so the variable of height being maintained will be regularly intervened by the policy variable which is number of times wings flapped. Hence for the total travel time of the bird this both variables might or might not have same mean value. Because bird might have a full tummy so it might have to intervene more regularly than normal or there is favorable wind which is causing less intervention. This property of the variable which can be intervened or can be used to intervene, makes the variable non random and in statistical terms it becomes non stationary.
Now what will happen if we do not incorporate this non random behavior into our regression model and we just estimate OLS which assumes that data is random or at-least stationary (there is a slight difference between them). Lets see what can be the issue here. we have dependent variable which have to be near to the target not less and not more and we have independent variable which is successfully keeping it on the target on average (I said successfully because I am assuming that bird has reached to the target, in statistical terms it means the process is not biased means bird exactly know how much intervention is needed), the problem here is that even though the average high needed to be maintained is a constant but it does not mean that bird will not form a sine wave around it might go down a little if there is too cold weather or it might go high a little if there are mountains in avoiding hunting. so if the dependent is forming a sort of sine wave among the constant target, residuals will also make a sort of inverted sine wave around its mean value of zero. since the sine wave forming dependent variable and independent variable which is also a sign wave by it is trying to dampen the wave formation of dependent variable both are non stationary means both are predictable from recent past hence definitely residuals will also become predictable and non random(will be seen by less than 2 value of Durban Watson).
Here the regression is apparently perfect the T values we will find will be way beyond the threshold of 2. The question here will be that if we know that the residuals are predictable then there must be other factors which are assisting the flight if we do not include them then there is no way to tell if the coefficient of wings flapped is valid, it may or may not be the necessary variable. This is creating a doubt on the regression results from the time behavior of the variables, if this doubt is correct that wings flapped are not relevant intervention then this regression will be called spurious regression. It is called spurious regression because we know that our selected variables are theoretically correct and this regression apparently confirms it too.
In order to validate the results we need to check one behavior if is exists or not. That behavior is correct implementation of intervention. This means when ever bird is going down (means there is negative error deviation from target) the intervention process kicks in and increase the height of the bird such that the change in height becomes negatively related to the error. Similarly if bird gains more height (means there is positive error deviation from target) then the intervention process reduces the height again proving that the change in height is negatively related to the error. This aspect can be seen from another angle note that when birds goes down the error term will have value less than zero mean and because of intervention the bird will go up which will make the residuals more than zero and consequently errors will be cancelled out confirming that future error is negatively related to past error. These two are explaining one aspect using two different ways and this is called cointegration or error correction, and in economics or applied terms it is called equilibrium as two forces i.e. height maintenance and the wings being flapped are being balanced forming zero mean residuals.
Δ Hight maintained = – f (error in past instance)
That is why when we use non stationary variable we usually make Error Correction Model in which we have change of dependent as dependent variable some lags of change of independent variables and then the past error term.
Now people as why should the coefficient of past error term must be between 0 and -1? since we have already explained why it should be negative. The answer is if there is any negative error in the past bird only have to create exactly equal and opposite error in future to correct it in ideal case, but it can make slightly less positive and opposite error and remaining can be compensated in coming future time. The smaller the time taken to correct the past error shows fast converging behavior which shows strength of cointegration. Now suppose bird makes positive error in future but it is higher in magnitude then it means bird will gain more height then the target height will require another intervention in reverse to bring it down (extra effort means less efficient process and more time taken to maintain the height).
Now the last aspect is the role of lag values of the first differences of dependent and independent variable in the short run equation. The lag of first difference of dependent will actually represent the inertia in the change process, like if the bird is too light and a small movement up or down also transfers into the movement in future. and the lag of first difference of independent intervening variable shows the aftereffects of the intervention if the wings are too strong and big slight intervention will affect present movement and future movement also. Hence we use appropriate lag order which is significant in the model in order to correctly calculate the behavior. So final equation becomes
heightt = α + β (no of wings flapped)t + εt
Δ heightt = θ + Σi = 1 n [ρi Δ heightt-i] + Σj = 0 n [ηj Δ no of wings flappedt-j] – πεt-1
The first equation is showing the behavior of horizon means that it is approximation of total journey from start to end what can be observed by the person who is looking at the bird from horizon. this is called long run equation. Second equation is showing minute temporary changes in the flight path which will eventually nullify at the end of the journey, that is why this is called short run equation.